# Tagged Questions

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### Proving the arithmetic mean equals the geometric mean when $a=b$.

Arithmetic mean $a,b \in \mathbb R$ is $A(a,b)=\frac{a+b}{2}$ Geomtric mean $a,b \in\left[0,\infty\right]$ is $G(a,b)=\sqrt{ab}$ I'm trying to prove that $G(a,b)=A(a,b)$ if and only if $a=b$. ...
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### If $(y_n(x))_{n \in \mathbb{N}}$ is uniformly convergent, so is $(f(x,y_n(x)))_{n \in \mathbb{N}} \ ?$

Let $f$ be a continuous function defined on $[a,b] \times [c,d].$ Consider $(y_n(x))_{n \in \mathbb{N}}$ such that it is uniformly convergent on $[e,f] \subseteq [a,b].$ Could anyone advise me on ...
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### If a continuous function is positive at a point, it is also positive in some neighborhood of the point [closed]

Suppose that $f:\mathbb{R}^k\to\mathbb{R}^1$ is a continuous function and that $f(x^*)>0$. Show that there is a ball $B=B_\delta(x^*)$ such that $f(x)>0$ for all $x\in B$.
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### Convergent or divergent $\sum_{n=1}^{\infty} \frac{e^nn!}{n^n}$?

Any suggestion/hint, not the whole solution, how to determine convergence/divergence of $$\sum_{n=1}^{\infty}\dfrac{e^n \cdot n!}{n^n}$$ I'm currently stuck.
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### Giving restriction on the value of $N$ in $\epsilon-N$ proof.

I'm still working on $\epsilon-N$ proof. If you don't mind is it possible for us to give restriction on the value of $N$ as illustrated by this example: Say after some manipulation of the limit ...
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### Choosing the right N in $\epsilon-N$ proof

I'm just a little bit confused in choosing the right $N$ when working on the rough sketch of the proof. Suppose after some algebra we have reached the point where we get this expression, say: ...
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### Alternative proof for the fact that a continuous function on a closed interval attains its boundaries.

Let $f:[a,b]\to \mathbb{R}$ be a continuous function. We are interested in showing that $\exists \beta \in [a,b]$, such that $f(\beta) = M$, where M is its upper boundary. I have managed to proof ...
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### Considering $\epsilon$ intuitively in limit proof

I'm having rather difficult time in trying to use $\epsilon$ argument appropriately. For example here is my simple $\epsilon$ proof in one question. The question is as follow: Prove if $s_n \geq 0$ ...
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### Intro to Real Analysis

I am having trouble proving the following: if $a < b$, then $a < {a+b\over2} < b$. I started with the Trichotomy Property and getting to where $a^2>0$, but then I do not know where ...
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### Question about the density of Q in R

So I was looking over a density that shows that the rational numbers are dense in the real numbers. If $0< a <b$, with with $a,b$ real numbers, then I understood why we can chose n such that: ...
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### U-substitution proof by partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $C^1$ on $(c,d)$. Then define ...
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### Show a double-sided infinite integral of $\sin(x+b)$ exists iff $b=n\pi$

More formally: Show that $$\lim_{a\rightarrow \infty} \int_{-a}^a \sin(x+b)$$ exists if and only if $b=n\pi$ for some $n \in \mathbb{Z}$. I get the intuition fine. The function is just a horizontal ...
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### Proving u-substitution the hard way — use only definition of integration with partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $C^1$ on $(c,d)$. Then define ...
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### Ideas on how to proceed with a proof?

Sorry if this is a nonspecific question - I can provide more details but at this point I need general ideas on a proof strategy. So I recently reduced a rather difficult optimization problem to ...
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### Passing a derivative through a limit.

After searching around on the net and on SE I have not found a satisfactory answer. Let $f_n: D \to \mathbb R$ be a sequence of functions. What assumptions, aside from $f$ being differentiable, do we ...
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### Single Variable Integral

Compute the following integral \int_0^{\Large\frac{\pi}{4}}\left[\frac{(1-x^2)\ln(1+x^2)+(1+x^2)-(1-x^2)\ln(1-x^2)}{(1-x^4)(1+x^2)}\right] x\, ...
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### Strategy to solve inequalities

I want to prove $|x + y| = |x| + |y|$ iff $xy \ge 0$. I don't understand a place to start. I was thinking of solving using contradiction but, as I am new to real analysis, I don't understand it. ...
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### Show if this is integrable (defined 1 on rationals, 0 else)

Define $f: [0,1] \rightarrow \mathbb{R}$ as $f(x) = \begin{cases} 0 & x \in \mathbb{Q} \\ 1 & x \notin \mathbb{Q} \end{cases}$ Find $\underline{\int_0^1f}$ and $\overline{\int_0^1f}$. Is ...
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### We are interested in price of a commodity, traded at regular intervals. Why it is reasonable to take $a$, $c$, and $d > 0$ and $b < 0$?

We are interested in the price of a commodity which is traded at regular intervals. We let $Q_k$ denote the supply of commodity, $D_k$ the demand for the commodity, and $p_k$ the price at $k$-th time. ...
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### A tough one: show that this is not differentiable at any point in R

Here's the question: Define $\phi: \ \mathbb{R} \rightarrow \mathbb{R}$ by $$\phi(x) = \begin{cases}x & 0\leq x\leq\frac{1}{2}\\ 1-x & \frac{1}{2}\leq x\leq 1\end{cases}.$$ And then ...
### show that $f(x,y) =2x^2 + 3y$ is differentiable at $(0,0)$ by finding a linear function T
Here's the question: Prove that $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined $f(x,y) = 2x^2 + 3y$ is differentiable at $\begin{bmatrix} 0\\0 \end{bmatrix}$ by producing a linear function T and ...
### Prove that the set of triples $\{(a,b,c)|a,b,c \in \mathbb{N}\}$ is countable
I have the following question in my textbook: Prove that the set of triples $\{(a,b,c)|a,b,c \in \mathbb{N}\}$ is countable Now I know that $\mathbb{N}$ is countable already, and I have completed a ...